Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs

A total dominating set in a graph is a set of vertices such that every vertex of the graph has a neighbor in the set. We introduce and study graphs that admit non-negative real weights associated to their vertices such that a set of vertices is a total dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We show that these graphs, which we call total domishold graphs, form a non-hereditary class of graphs properly containing the classes of threshold graphs and the complements of domishold graphs, and are closely related to threshold Boolean functions and threshold hypergraphs. We present a polynomial time recognition algorithm of total domishold graphs, and characterize graphs in which the above property holds in a hereditary sense. Our characterization is obtained by studying a new family of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of independent interest.Comment: 19 pages, 1 figur

Fair allocation of indivisible goods under conflict constraints

We consider the fair allocation of indivisible items to several agents and add a graph theoretical perspective to this classical problem. Thereby we introduce an incompatibility relation between pairs of items described in terms of a conflict graph. Every subset of items assigned to one agent has to form an independent set in this graph. Thus, the allocation of items to the agents corresponds to a partial coloring of the conflict graph. Every agent has its own profit valuation for every item. Aiming at a fair allocation, our goal is the maximization of the lowest total profit of items allocated to any one of the agents. The resulting optimization problem contains, as special cases, both {\sc Partition} and {\sc Independent Set}. In our contribution we derive complexity and algorithmic results depending on the properties of the given graph. We can show that the problem is strongly NP-hard for bipartite graphs and their line graphs, and solvable in pseudo-polynomial time for the classes of chordal graphs, cocomparability graphs, biconvex bipartite graphs, and graphs of bounded treewidth. Each of the pseudo-polynomial algorithms can also be turned into a fully polynomial approximation scheme (FPTAS).Comment: A preliminary version containing some of the results presented here appeared in the proceedings of IWOCA 2020. Version 3 contains an appendix with a remark about biconvex bipartite graph

Guidelines for the use and interpretation of assays for monitoring autophagy (3rd edition)

In 2008 we published the first set of guidelines for standardizing research in autophagy. Since then, research on this topic has continued to accelerate, and many new scientists have entered the field. Our knowledge base and relevant new technologies have also been expanding. Accordingly, it is important to update these guidelines for monitoring autophagy in different organisms. Various reviews have described the range of assays that have been used for this purpose. Nevertheless, there continues to be confusion regarding acceptable methods to measure autophagy, especially in multicellular eukaryotes. For example, a key point that needs to be emphasized is that there is a difference between measurements that monitor the numbers or volume of autophagic elements (e.g., autophagosomes or autolysosomes) at any stage of the autophagic process versus those that measure fl ux through the autophagy pathway (i.e., the complete process including the amount and rate of cargo sequestered and degraded). In particular, a block in macroautophagy that results in autophagosome accumulation must be differentiated from stimuli that increase autophagic activity, defi ned as increased autophagy induction coupled with increased delivery to, and degradation within, lysosomes (inmost higher eukaryotes and some protists such as Dictyostelium ) or the vacuole (in plants and fungi). In other words, it is especially important that investigators new to the fi eld understand that the appearance of more autophagosomes does not necessarily equate with more autophagy. In fact, in many cases, autophagosomes accumulate because of a block in trafficking to lysosomes without a concomitant change in autophagosome biogenesis, whereas an increase in autolysosomes may reflect a reduction in degradative activity. It is worth emphasizing here that lysosomal digestion is a stage of autophagy and evaluating its competence is a crucial part of the evaluation of autophagic flux, or complete autophagy. Here, we present a set of guidelines for the selection and interpretation of methods for use by investigators who aim to examine macroautophagy and related processes, as well as for reviewers who need to provide realistic and reasonable critiques of papers that are focused on these processes. These guidelines are not meant to be a formulaic set of rules, because the appropriate assays depend in part on the question being asked and the system being used. In addition, we emphasize that no individual assay is guaranteed to be the most appropriate one in every situation, and we strongly recommend the use of multiple assays to monitor autophagy. Along these lines, because of the potential for pleiotropic effects due to blocking autophagy through genetic manipulation it is imperative to delete or knock down more than one autophagy-related gene. In addition, some individual Atg proteins, or groups of proteins, are involved in other cellular pathways so not all Atg proteins can be used as a specific marker for an autophagic process. In these guidelines, we consider these various methods of assessing autophagy and what information can, or cannot, be obtained from them. Finally, by discussing the merits and limits of particular autophagy assays, we hope to encourage technical innovation in the field

Linearna separacija povezanih dominantnih množic v grafih

Povezana dominantna množica v grafu je dominantna množica točk, ki inducira povezan podgraf. Po zgledu sorodnih raziskav v literaturi o neodvisnih množicah, dominantnih množicah in totalno dominantnih množicah v tem članku raziskujemo razred grafov, v katerem lahko povezane dominantne množice točk ločimo od ostalih podmnožic točk z linearno utežno funkcijo. Natančneje, pravimo, da je graf povezano dominantno pragoven, če lahko njegovi množici točk priredimo take nenegativne realne uteži, da je množica točk povezana dominantna množica natanko tedaj, ko vsota uteži njenih elementov preseže določen prag. Grafe tega nehereditarnega razreda karakteriziramo s pomočjo množice minimalnih prerezov grafa. Podamo tudi več karakterizacij za hereditarni primer, tj. ko se za vsak povezan induciran podgraf zahteva, da je povezano dominantno pragoven. Karakterizacija s prepovedanimi induciranimi podgrafi implicira, da je ta razred grafov prava posplošitev dobro znanih razredov tetivnih grafov, bločnih grafov in trivialno popolnih grafov. Preučujemo tudi določene algoritmične vidike povezano dominantno pragovnih grafov. Na podlagi povezav z minimalnimi prerezi in lastnostmi izpeljanih hipergrafov in Boolovih funkcij pokažemo, da naš pristop vodi k novim polinomsko rešljivim primerom problema utežene povezane dominantne množice

On matching extendability of lexicographic products

A graph G of even order is ℓ-extendable if it is of order at least 2ℓ + 2, contains a matching of size ℓ, and if every such matching is contained in a perfect matching of G. In this paper, we study the extendability of lexicographic products of graphs. We characterize graphs G and H such that their lexicographic product is not 1-extendable. We also provide several conditions on the graphs G and H under which their lexicographic product is 2-extendable